Obtaining current list of companies in the FTSE 100 via an API

Most index distributors publish current holdings for their indexes but historical holdings are only generally available for a cost. Data distribution is the primary way index providers (eg, FTSE Russell in this case) make their money, they're not giving away this stuff for free.

The London Stock Exchange maintains a public list of current FTSE 100 constituents; however, getting historical components (much less weights) is more difficult.

You can look at revisions to the wikipedia page to get some of the changes; however, for good data on historical index composition you will need to pay money.

Finally, one note: indices do not always contain their full number of constituents. When a firm goes bankrupt, is delisted, or merges with another index constituent, the index temporarily has fewer firms. That gets fixed after the index maintainers decide which firm will replace those which left the index.

You could use the Bloomberg BEQS function as part of the API, but you need to have a Bloomberg Terminal subscription.